
Assumption REF  i. implies that \(\Vert \Psi (\hat{\theta }_n)^{b}(\theta _0-\hat{\theta }_n)\Vert _2 \le \overline{\rho }^{b}\Vert \theta _0-\hat{\theta }_n\Vert _2\) , so the first term is less than \(\frac{d_{0,n}}{(1-\overline{\rho })B}\)  in expectation. Consider now the second term. Using the relationship between \(\Vert \cdot \Vert _2\)  and the Frobenius \(\Vert \cdot \Vert _F\)  norms, [1]}, the zero-mean condition (Assumption REF  i.), and the conditional iid assumption, we have:
\(\mathbb {E}^\star \left( \Vert \frac{1}{B} \sum _{b=1}^B \sum _{j=0}^{b-1} \Psi _n^j\overline{P}_m G_m^{(b-j)}(\hat{\theta }_n)\Vert _2^2 \right) &\le \mathbb {E}^\star \left( \Vert \frac{1}{B} \sum _{b=1}^B \sum _{j=0}^{b-1} \Psi _n^j\overline{P}_m G_m^{(b-j)}(\hat{\theta }_n)\Vert _F^2 \right)\\& = \text{trace}\left( \text{var}^\star \left[ \frac{1}{B} \sum _{b=1}^B \sum _{j=0}^{b-1} \Psi _n^j\overline{P}_m G_m^{(b-j)}(\hat{\theta }_n) \right] \right)\\&= \frac{1}{B^2} \sum _{b=1}^B \text{trace}\left( \text{var}^\star \left[ \sum _{j=0}^{B-b} \Psi _n^{j} \overline{P}_m G_m^{(b)} \right] \right)\\&= \frac{1}{B^2 m} \sum _{b=1}^B \text{trace}\left( \left[\sum _{j=0}^{B-b}\Psi _n^{j} \right] \overline{P}_m \Sigma _n \overline{P}_m^\prime \left[ \sum _{j=0}^{B-b}\Psi _n^{j} \right]^\prime \right)\\&\le \frac{\overline{\lambda }_P^2 \text{trace}(\Sigma _n)}{(1-\overline{\rho })^2} \frac{1}{mB}, \) 
